No. Whether you're a citizen, a resident alien, or an illegal immigrant, you're free to leave the country so long as there's no warrant out for your arrest. The United States doesn't require exit visas or have a comprehensive exit-control system, and southbound travelers are more likely to encounter Mexican customs agents than American law enforcement officers. According to Department of Homeland Security data, as many as 100,000 undocumented workers have left Arizona in the past two years, either to return home or try out other states. Although Americans for Legal Immigration promotes "safe departure" checkpoints as a way to eliminate "desert crossings, [or] paying money to the cartels for passage south," such risks aren't necessary. Migrants can simply use public transportation, as detailed in this CNN story about an illegal immigrant from Guatemala who boarded a bus with a one-way ticket, paid for by the Guatemalan consulate.[More]

Of course they're missing one point—the government could arrest people they know to have entered illegally, and if they could prove illegal entry, or working illegally, then they could charge the illegals with the relevant Federal crimes. They just choose not to.